Graceful Labeling for Some Star Related Graphs

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1 International Mathematical Forum, Vol. 9, 2014, no. 26, HIKARI Ltd, Graceful Labeling for Some Star Related Graphs V. J. Kaneria, M. M. Jariya and Meera Meghpara Department of Mathematics, Saurashtra University Rajkot , India Copyright c 2014 V. J. Kaneria, M. M. Jariya and Meera Meghpara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We have investigate graceful labeling and cordial labeling for the graphs obtained by joining apex vertices of some star graphs to a new vertex. Mathematics Subject Classification: 05C78 Keywords : Graceful labeling, Star, Cordial labeling 1 Introduction : We begin with the simple, connected, undirected and finite graph G = (V, E). For all standard terminology and notations we follow Harary [4]. We will give brief summery of definitions which are used in this paper. Definition 1.1: If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.

2 1290 V. J. Kaneria, M. M. Jariya and Meera Meghpara Definition 1.2: Given a graph G = (V, E), the set N of non negative integers and a commutative binary operation : N N N, every vertex f : V N induces an edge function f : E N such that f (uv) = f(u) f(v), for all uv E. A function f is called graceful labeling of a graph G if f : V {0, 1, 2,..., q} is injective and the induced function f : E {1, 2,..., q} defined as f (e = uv) = f(u) f(v) is bijective. A graph which admits graceful labeling is called graceful graph. Definition 1.3: A function f : V {0, 1} is called binary vertex labeling of a graph G and f(v) is called label of the vertex v of G under f. For an edge e = uv, the induced function f : E {0, 1} is given as f (e = uv) = f(u) f(v). Let v f (0), v f (1) be number of vertices of G having labels 0 and 1 respectively under f and let e f (0), e f (1) be number of edges of G having labels 0 and 1 respectively under f. A binary vertex labeling f of a graph G is called cordial labeling if v f (0) v f (1) 1 and e f (0) e f (1) 1. A graph which admits cordial labeling is called cordial graph. Definition 1.4: Let G = (V, E) be a graph. A mapping f : V {0,1,2} is called ternary vertex labeling of G and f(v) is called label of the vertex v of G under f. For an edge e = uv the induced edge labeling f : E {0,1,2} is given by f (e) = f(u) f(v). Let v f (0),v f (1),v f (2) be the number of vertices of G having labels 0,1 and 2 respectively under f and let e f (0),e f (1),e f (2) be the number of edges having labels 0,1 and 2 respectively under f. A ternary vertex labeling of a graph G is called 3 equitable labeling if v f (i) v f (j) 1 and e f (i) e f (j) 1, 0 i, j 2. A graph G is called 3 equitable graph if it admits 3 equitable labeling. Definition 1.5: Consider t copies of stars namely K 1,n1, K 1,n2,..., K 1,nt. Then the G =< K 1,n1, K 1,n2,..., K 1,nt > is the graph obtained by joining apex vertices of each K 1,ni and K 1,ni+1 to a new vertex u i, where 1 i t 1. For detail survey of graph labeling one can refer Gallian [3]. Vast amount of literature is available on different type of graph labeling. Labeled graphs have variety of applications in coding theory particularly missile guidance codes. A detail study of variety of applications of graph labeling is given by bloom and Golomb [1].

3 Graceful labeling for some star related graphs 1291 The graceful labeling was introduced by A. Rosa [5] during He proved that the cycle C n is graceful, when n 0, 3 (mod 4). Rosa [5] and Golomb [2] proved that the complete graph bipartite graphs K m,n are graceful. Vaidya et al. [6] proved a graph obtained by some copies of star by joining their apex with a new vertex is cordial as well as 3 equitable. 2 Main Results : Theorem 2.1 : The graph G obtained by joining some stars < K 1,n1, K 1,n2,..., K 1,nt > is graceful. Proof : Let v i,0, v i,1,..., v i,ni i = 1, 2,..., t be vertices of star graphs K 1,ni, i = 1, 2,..., t. We shall join these graphs K 1,ni and K 1,ni+1 by new vertex u i, where 1 i t 1 by their apex vertices. We define labeling function f : V {0, 1,..., q}, where q = Σ t i=1 n i +(2t 2) as follows: G. f(v i,0 ) = i 1, i = 1, 2,..., t. f(v 1,j ) = q (j 1), j = 1, 2,..., n 1. f(u j ) = q Σ j l=1 n l (j 1) f(v j,i ) = q Σ j 1 l=1 n l (j + i 2), i = 1, 2,..., n j ; j = 2, 3,..., t. The above labeling pattern give rise graceful labeling to the given graph Illustration 2.2: A graph G = < K 1,6, K 1,5, K 1,3 > and its graceful labeling is shown in following Figure 1. Figure 1

4 1292 V. J. Kaneria, M. M. Jariya and Meera Meghpara Let v f (odd), v f (even) be the number of vertices of a graceful graph G with labeling function f having odd labels and even labels respectively under f. Also e f (even)= the number of even edge label of the graph G, e f (odd)= the number of odd edge label of the graph G under the corresponding function f. We know that for a graceful graph G with graceful labeling f, e f (odd) e f (even) 1. This property will help for following result. Theorem 2.3 : For a graceful graph G with graceful labeling f, if it satisfies the condition v f (odd) v f (even) 1 then G is a cordial graph. Proof : We define g : V {0, 1}, the binary vertex labeling for G as follows: f(u) = 1 if u has odd vertex label and f(u) = 0 if u has even vertex label under graceful labeling f. Then we see that v g (0) v g (1) 1 and e g (0) e g (1) 1, because e g (0) = e f (even) and e g (1) = e f (odd), where v g (0), v g (1) be number of vertices of G having labels 0 and 1 respectively under g and let e g (0), e g (1) be number of edges of G having labels 0 and 1 respectively under the induced function g by g. Therefore G admits a cordial labeling g and so G is a cordial graph. Illustration 2.4: K 4,3 with its graceful labeling and the corresponding cordial labeling is shown in following Figure 2. Figure 2 Remarks : Every graceful graph does not have the property v f (odd) v f (even) 1, where f is graceful labeling for G and so every graceful graph may not be cordial. We know that K 4 is graceful graph, while it is not a cordial graph as in K 4, v f (odd)= 1 and v f (even)= 3.

5 Graceful labeling for some star related graphs 1293 Corollary 2.5 : The graph G = < K 1,n1, K 1,n2,..., K 1,nt > is cordial. Proof : Above graph G is a tree and so the number of vertices for G is q + 1, where q = the number of edges for G, which is here Σ t i=1 n i + 2(t 1). So the graceful labeling f for G is bijective map. Hence it satisfies v f (odd) v f (even) 1. Therefore by Theorem 2.3 G is a cordial graph. 3 Concluding Remarks : Labeled graph is the topics of current interest. We discussed here graceful labeling and cordial labeling of some star related graphs. In [6] vaidya et al. proved that the graph obtained by < K1,n; 1 K1,n; 2... ; K1,n k > (k copies of K 1,n ) is cordial graph, which is particular case of our Corollary 2.5 by taking n 1 = n 2 =... = n t = n and t = k. References [1] G. S. Bloom and S. W. Golomb, Application of numbered undirected graphs, Proceedings of IEEE, 165 (4) (1977), [2] G. S. Bloom and S. W. Golomb, Numbered complete graphs, usual rules and assorted applications, in Theory and applications of Graphs, Lecture Notes in Math., 642, Springer Verlag, New York (1978) [3] J. A. Gallian, The Electronics Journal of Combinatorics, 16, DS6(2013). [4] F. Harary, Graph theory Addition Wesley, Massachusetts, [5] A. Rosa, On certain valuation of graph, Theory of Graphs (Rome, July 1966), Goden and Breach, N. Y. and Paris, 1967, [6] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, Cordial and 3 equitable labeling for some Star related graphs, International Mathematical Forum, 4, (2009) Received: April 15, 2014